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Real-time evolution of replicas of classical field is proposed as anapproximate simulator of real-time quantum field dynamics at finitetemperatures. We consider $N$ classical field configurations dubbed as replicaswhich interact with each other via the $\tau$-derivative terms and evolve withthe classical equation of motion. The partition function of replicas is foundto be proportional to that of quantum field in the imaginary time formalism. Asthe replica index $\tau$ can be regarded as the imaginary time index, thereplica evolution is technically the same as the molecular dynamics part of thehybrid Monte-Carlo sampling and the replica configurations should reproduce thecorrect quantum equilibrium distribution after the long-time evolution. At thesame time, evolution of the replica-index average of field variables isdescribed by the classical equation of motion when the fluctuations are small.In order to examine the real-time propagation properties of replicas, we firstdiscuss replica evolution in quantum mechanics. Statistical averages ofobservables are precisely obtained by the initial condition average of replicaevolution, and the time evolution of the unequal-time correlation function,$\langle x(t) x(t')\rangle$, in a harmonic oscillator is also described well bythe replica evolution in the range $T/\omega > 0.5$. Next, we examine thestatistical and dynamical properties of the $\phi^4$ theory in the 4+1dimensional spacetime, which contains three spatial, one replica index or theimaginary time, and one real-time. We note that the Rayleigh-Jeans divergencecan be removed in replica evolution with $N \geq 2$ when the mass countertermis taken into account. We also find that the thermal mass obtained from theunequal-time correlation function at zero momentum grows as a function of thecoupling as in the perturbative estimate in the small coupling region.

Replica evolution of classical field in 4+1 dimensional spacetime toward real time dynamics of quantum field